Locally bounded topologies on Euclidean rings

Abstract

0. Let (A, J ) be a topological ring. A subset B c A is bounded if for every neighbourhood U of zero there exists a neighbourhood V of zero such that B V r U and VB c U. A topology ~on A is locally bounded if there exists a bounded neighbonrhood of zero. A topological ring (A, J ) is precompact if the completion _~ of A at 3 is a compact topological ring. Mutylin [5] has remarked tha t every locally bounded Hausdorff nondiserete ring topology of • has an ideal base of zero noighbourhoods. To prove it, it is sufficient to show tha t for every such topology on F, every neighbourhood of zero contains a nonzero ideal. For other proofs of this fact see [1] and [6]. In this note we find a class of Euclidean rings with the property tha t every nondiscrete locally bounded Hausdorff ring topology is precompact, i.e. has an ideal base of neighbourhoods of zero.